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Pronormality of Hall Subgroups in Their Normal Closure. / Vdovin, E. P.; Nesterov, M. N.; Revin, D. O.

In: Algebra and Logic, Vol. 56, No. 6, 01.01.2018, p. 451-457.

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Vdovin EP, Nesterov MN, Revin DO. Pronormality of Hall Subgroups in Their Normal Closure. Algebra and Logic. 2018 Jan 1;56(6):451-457. doi: 10.1007/s10469-018-9467-8

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Vdovin, E. P. ; Nesterov, M. N. ; Revin, D. O. / Pronormality of Hall Subgroups in Their Normal Closure. In: Algebra and Logic. 2018 ; Vol. 56, No. 6. pp. 451-457.

BibTeX

@article{58b48d60da384ae99b20b93a5490982d,
title = "Pronormality of Hall Subgroups in Their Normal Closure",
abstract = "It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.",
keywords = "normal closure, pronormal subgroup, π-Hall subgroup, pi-Hall subgroup",
author = "Vdovin, {E. P.} and Nesterov, {M. N.} and Revin, {D. O.}",
year = "2018",
month = jan,
day = "1",
doi = "10.1007/s10469-018-9467-8",
language = "English",
volume = "56",
pages = "451--457",
journal = "Algebra and Logic",
issn = "0002-5232",
publisher = "Springer US",
number = "6",

}

RIS

TY - JOUR

T1 - Pronormality of Hall Subgroups in Their Normal Closure

AU - Vdovin, E. P.

AU - Nesterov, M. N.

AU - Revin, D. O.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.

AB - It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.

KW - normal closure

KW - pronormal subgroup

KW - π-Hall subgroup

KW - pi-Hall subgroup

UR - http://www.scopus.com/inward/record.url?scp=85042436357&partnerID=8YFLogxK

U2 - 10.1007/s10469-018-9467-8

DO - 10.1007/s10469-018-9467-8

M3 - Article

AN - SCOPUS:85042436357

VL - 56

SP - 451

EP - 457

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 6

ER -

ID: 10428358